[plxztbea] Pierpont primes with primitive root 2

Pierpont primes have the form 1 + 2^a * 3^b.  We restrict our attention to the subset of Pierpont primes for which 2 is a generator of its multiplicative group.

First, consider a further subset in which the exponent of 2 is greater than 1, that is, a>1, and 2 is a generator.  Are the following primes the only members of this subset?  (Searched all Pierpont primes less than 2^10000.)  (Also checked 1 + 4*3^b up to b=100000.)

1 + 2^2 * 3^0 = 5
1 + 2^2 * 3^1 = 13
1 + 2^2 * 3^2 = 37

Next, we consider a=1.  Here are the first few b such 1 + 2*3^b is prime with primitive root 2.  (Searched up to b=68949.)

0 2 4 16 320 782 1252 1454 5480 64822

This sequence is a subset of OEIS A003306.

Pari/GP:

is(n)=my(x=1+2*3^n);ispseudoprime(x)&&(2==znprimroot(x))
for(n=0,+oo,if(is(n),print(n)))

More fancy, demonstrating parallel processing:

my(chunksize=10) ; parfor(c=0 , +oo , my(start=c*chunksize) ; my(end=start+chunksize-1) ; for(n=start , end , my(p=1+2*3^n) ; my(myisprime=1) ; forprime(t=5 , 1000000 , if(t*t>=p , break) ; if(p%t==0 , myisprime=0 ; break)) ; if(myisprime , my(y=Mod(2,p)^3^(n-1)) ; if(1==y^6 , if(y^3!=1 && y^2!=1 , print("found ",n))))) ; print("# ",end))